A Theory of Photometric Reconstruction for Unknown Isotropic Reflectances

This paper presents a comprehensive theory of surface reconstruction from image derivatives in photometric stereo. For an object with an unknown, general isotropic BRDF, we show that just two measurements of the spatial and temporal image derivatives, under unknown light source positions on a circle, suffice to determine the surface. This result is the culmination of a series of fundamental observations. First, we discover a photometric invariant that relates image derivatives to the surface geometry, regardless of the form of isotropic BRDF. Next, we show that just two pairs of differential images from unknown light directions suffice to recover this surface information from the photometric invariant. This is shown to be equivalent to determining isocontours of constant magnitude of the surface gradient, as well as isocontours of constant depth, for the entire surface. Further, we prove that specification of the surface normal at a single point completely determines the surface depth from these isocontours. In addition, our theory also suggests practical algorithms that require additional initial or boundary information, but allow reconstruction of depth from lower order derivatives. The theoretical results of the paper are illustrated with examples on synthetic and real data.